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Full text of "Mathematical And Physical Papers - Iii"

44     ON THE EFFECT  OF THE  INTERNAL FRICTION  OF FLUIDS
If now we put X = XV"1, we shall have 0 and \r for the limits of X', and the second of these becomes infinite with r.   Hence
limit of F,(r) = (2r)-* e f \C" + D" log 2V) e-*' X'^ (ZX' ...... (91).
Jo
r
Now     e"* af * dx = TT*, and if we differentiate both sides of the '
[  e-xaf-*dx = T(s)
Jo
with respect to s, and after differentiation put s = , we get f   e"*fl?-*logacte
JO
Putting x = mX; in these equations we get
'o equation
r
Jo
f *e-x' x'-* log V d\' = m-* {F (|) - TT*log m}, Jo
where that value of m~* is to be taken which has its real part positive.    Substituting in (91) we get
limit of FJr) = (^-V emr { C" + (TT-* T'| - log ~") D"} . 3V ;     \2<mrJ       \         \           2        6 2J      j
Comparing with (88) we get
30.   We are now enabled to find the relation between C and
jD arising from the condition that the motion of the fluid shall not
become infinitely great at an infinite distance from the cylinder.
The determination of the arbitrary constants A, B,  C, D will
present no further difficulty,    We must have 5 = 0, since other-
wise the velocity would be finite at an infinite distance, and then
the two equations (83), combined with the relation above men-
tioned, will serve to determine A, (7, D,   The motion of the fluid
will thus be completely determined, the functions F^(r)y  F8(r)
being given by (84) and (87).   When the modulus of mr is large,
the series in (87), though ultimately hypergeometrically conver-